Bouncing Wave Packets, Ehrenfest Theorem, and Uncertainty Relation based upon a new Concept for the Momentum of a Particle in a Box

Abstract

For a particle in a box, the operator $-i\partial_x$ is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg-Robertson-Schr\"odinger uncertainty relation and show that our construction also provides a physical interpretation for it.